(see eq:unif). Notice that in the case when \(\mathcal {N}(\varepsilon , {H}, \mathsf {d}_{{H}}) = \mathrm {e}^{C\varepsilon ^{-2d / \gamma }}\)  (see eq:ex) the sample size (in the above relation or TM1.1) is of order \(\varepsilon ^{-(2 + 2d / \gamma )}\) . On the oder hand, one expects that the sample size \(n\)  needed to ensure that \((1 + \alpha )\hat{\rm er}_n(h) + \varepsilon \ge \mathrm {er}_{}(h)\)  for some \(\alpha , \varepsilon > 0\)  is of much smaller order. This problem
was studied in [1]}. Crucial step in the proof of this result is to study the uniform relative error
\(\sup _{h \in {H}} \frac{\left| \mathrm {er}_{}(h) - \hat{\rm er}_n(h) \right|}{\sqrt{\mathrm {er}_{}(h)}},\) 
